I found the tripadvisor Budapest forum so helpful for planning our trip that I thought I'd come back and write up a few highlights and tips. I advise others planning a trip to do a search on this forum for what you are interested in as it has a lot of info already here.
First of all there are free maps available everywhere when you arrive at the airport - these are very helpful and give clear instructions on how to use the Bus and Metro to get to the city centre.
TRANSPORT: There were just the two of us and we didn't have much luggage and we are used to finding our way around... so we went for the Bus and Metro option - so much cheaper.
Buy a bus ticket in the newsagent (shop) in the airport.. cost about 160ft each. When you leave the terminal - make sure you walk out to the main road to the bus stop(about 100metres) - bus stop is right there, bus 93 every 15 mins. You can't buy the 3 day travel pass (2500ft each) at the airport (maybe you can the Buda card, which is more expensive and only worth it if you are an avid museum goer) but you can buy every kind of ticket when you get to the metro station (Kobanya-Kispest). In fact, we worked out that we probably weren't going to use the travel pass enough to make it the most economical option since our hotel was v.central - so we bought a book of 20 tickets to share for 3145ft - and had all the fun of stamping our tickets. Make sure you do this BEFORE you get on the train (or as soon as you get on the bus/tram) in the little orange ticket machines!
Using public transport around the city is EASY - which makes it a great place to visit. All the bus, tram and metro lines are clearly marked on the free map.
HOTEL: We stayed at Le Meridien - which was FANTASTIC. Very conveniently placed at the juncture of the 3 metro lines (Deak ter.) and a lovely hotel. We got a great online deal - Â£70 a night plus tax and since it was our Anniversary they upgraded us and left a bottle of bubbly in our room. Lovely service, beautiful hotel, great rooms, great sauna/pool/spa included. But pricey breakfast - which wasn't include for us - but okay room service and cafes locally.
GOOD THINGS TO DO:
Central Market - indoor covered market at end of Vaci Utca (main shopping street). Great for traditional food, crafts. Go upstairs, on one side are little food stalls which obviously locals eat at... interesting hungarian food, glasses of beer and hot wine at decent prices (150ft for .33l) - we loved it.
Chain Bridge/Castle Hill - we got the bus over the bridge, got off first stop on Buda side and got Siklo up to Royal Palace. Great fun. Lovely views even though weather v.cold and misty! Fishermans Bastion - got to be done though a bit twee!
Szechenyi Baths, we took the yellow line (historical underground - lovely) up to Heros Square (Hosok Ter.) and walked up through the park. Baths fantastic. All ticket info in English, so easy. Change - go outside to the outdoor thermal pools - fantastic when it's cold outside. Then go in the door on the other side of the pool through to all the indoor pools and steam rooms. I think it was about 1800ft to get in and you get a refund after 2hrs (700ft) and 3hrs (400ft). We got one cabin to change in and you can use the free safe boxes to keep valuables in.
Gundel - we couldn't afford to have dinner at Gundel, the most famous restaurant in Budapest... but they do a great afternoon tea in their bar. Very swish and just the same price for coffee and cake as we found in any old cafe in the tourist/shopping areas. A great experience.
Ice-skating - we went ice-skating in the same park, just by Heros Sq. It's a vast outdoor rink made out of part of the lake in winter. Very fun - cost 800ft each, plus 500ft skate hire for an hour. Take photo id for each person (or 15,000ft each) as deposit for skates. When we were there, it opened in am until 1pm, closed 1-4pm and then open 4pm onwards. There were queues for evening session, so worth getting there early.
Opera House - this is worth it. We booked on line (use Jegymester - ie ticket master in hungarian!)it was very easy - there is a seat plan and you can choose your exact seat. Tickets become available about a month before the performance and you have to get in quick to get good, but ridiculously cheap tickets. The inside of the theatre is stunning - all gold and painted ceilings. We saw the Swan Lake ballet - a great experience.
We would have liked to have more time to find some cheap and authentic buffets and other places.. but it was cold, so we didn't walk too far from our hotel, quite an expensive area in the centre. The advice on this forum is right - it's hard to know what will be good and it's not always great value in tourist/shopping areas.
We loved CAFE KOR on Sas Utca, which was great value and obviously popular with locals and many regulars. It was heaving on a Thursday night... advise to book ahead. But great atmosphere and good food. They don't take plastic, but do take any currency. Our 2 course meal, with soft drinks and service was 8000ft and was great.
We also liked Leroy, next to St.Stephan's Basilica and also on Sas Utca - it had a funky interior. 3 courses there was about Â£35 for 2 with wine and coffee.
We heard cafes around the Jewish quarter are good, but didn't quite get there.
Weather was bitterly cold in February, but we took warm coats, gloves, hats, scarves - all essential and it didn't spoil our time at all.
People - we can report that we found people to be really friendly and helpful. Some on this forum didn't have the same experience and a friend who had lived in B'pest for 4 years told us it's really got better in the last couple of years. Our first experience of this was on our first metro ride, with suitcase and map out... an old lady, could speak a word of English did everything she could to help us, so sweet... she kept talking to us and acting out the way to go (although we already knew!) and then another lady on the other side started translating for us with bemusement at the older ladies over the top attempts to help! They were so friendly. We had other encounters like this.
Shopping - prices were higher than I expected for everything - so not the ideal shopping city. But some interesting things. There is a mall near Moscow Square and another by the North Station - okay if you want to look around. We got some nice sausages, some honey and other produce - which were the only things really cheaper than home.
So this has been a long report, but we had a great time in what is a great city. We'd love to go back with our kids because we think it would be really easy with them too.
Hope you enjoy your trip!